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Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

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Abstract

The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proportional to this distance. Since we cannot know the distance to integrality a priori, we prove that the norm of the Moore–Penrose generalized inverse of the matrix of coefficients is a key factor for bounding this distance, and then we propose a preconditioning method to reduce this norm using lattice reduction techniques. We also propose a set of valid box constraints that help accelerate the RRBB method. Our computational results show that the proposed preconditioning significantly reduces the size of the RRBB enumeration tree, that the preconditioning combined with the proposed set of box constraints can significantly reduce the computational time of RRBB, and that the resulting RRBB method can outperform the Schnorr and Eucher method, a widely used method for solving integer least squares problems, on some types of problem data.

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References

  1. Aardal, K., Eisenbrand, F.: The LLL algorithm and integer programming. In: Nguyen, P.Q., Vallée, B. (eds.) The LLL Algorithm, Information Security and Cryptography, pp. 293–314. Springer, Berlin (2010)

  2. Aardal, K., Heymann, F.: On the structure of reduced kernel lattice bases. In: Goemans, M., Correa, J. (eds.) Integer Programming and Combinatorial Optimization, volume 7801 of Lecture Notes in Computer Science, pp. 1–12. Springer, Berlin (2013)

  3. Aardal, K., Weismantel, R., Wolsey, L.A.: Non-standard approaches to integer programming. Discret. Appl. Math. 123, 5–74 (2002)

    Article  Google Scholar 

  4. Aardal, K., Wolsey, L.A.: Lattice based extended formulations for integer linear equality systems. Math. Program. 121, 337–352 (2010)

    Article  Google Scholar 

  5. Agrell, E., Eriksson, T., Vardy, A., Zeger, K.: Closest point search in lattices. IEEE Trans. Inf. Theory 48, 2201–2214 (2002)

    Article  Google Scholar 

  6. Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Proceedings of STOC, pp. 284–293 (1997)

  7. Ajtai, M., Kumar, R., Sivakumar, D.: Sampling short lattice vectors and the closest lattice vector problem. In: Proceedings of CCC, pp. 53–57 (2002)

  8. Babai, L.: On Lóvasz’ lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986)

    Article  Google Scholar 

  9. Bremmer, M.R.: Lattice Basis Reduction, an Introduction to the LLL Algorithm and Its Applications. CRC Press, Boca Raton (2012)

  10. Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Math. Program. 135, 369–395 (2012)

    Article  Google Scholar 

  11. Chang, X.-W., Golub, G.H.: Solving ellipsoid-constrained integer least squares problems. SIAM J. Matrix Anal. Appl. 31, 1071–1089 (2009)

    Article  Google Scholar 

  12. Chang, X.-W., Paige, C.C.: Euclidean distances and least squares problems for a given set of vectors. Appl. Numer. Math. 57, 1240–1244 (2007)

    Article  Google Scholar 

  13. Chang, X.-W., Wen, J., Xie, X.: Effects of the LLL reduction on the success probability of the Babai point and on the complexity of sphere decoding. IEEE Trans. Inf. Theory 59, 4915–4926 (2013)

    Article  Google Scholar 

  14. Chang, X.W., Xie, X., Zhou, T.: MILES: MATLAB package for solving Mixed Integer LEast Squares problems, Version 2.0, October 2011. http://www.cs.mcgill.ca/~chang/software.php

  15. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  Google Scholar 

  16. Eisenbrand, F.: Integer programming and algorithmic geometry of numbers. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 505–559. Springer, Berlin (2010)

  17. Fincke, U., Pohst, M.: A procedure for determining algebraic integers of given norm. Proc. RUROCAL 162, 194–202 (1983)

    Google Scholar 

  18. Hanrot, G., Pujol, X., Stehle, D.: Algorithms for the shortest and closest lattice vector problems. In: Proceedings of the IWCC, pp. 159–190 (2011)

  19. Hassibi, A., Boyd, S.: Integer parameter estimation in linear models with applications to GPS. IEEE Trans. Signal Process. 46, 2938–2952 (1998)

    Article  Google Scholar 

  20. IBM, ILOG CPLEX Optimization Studio: http://www-01.ibm.com/software/integration/optimization/cplex-optimization-studio/

  21. Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: Proceedings of the STOC, pp. 99–108 (1983)

  22. Kisialiou, M., Luo, Z.Q.: Performance analysis of quasi-maximumlikelihood detector based on semi-definite programming. In: Proceedings of the IEEE ICASSP, pp. 433–436 (2005)

  23. Krishnamoorthy, B., Pataki, G.: Column basis reduction and decomposable knapsack problems. Discret. Optim. 6, 242–270 (2009)

    Article  Google Scholar 

  24. Ku, W.Y.: Lattice Preconditioning for the Real Relaxation Based Branch and Bound Method for Integer Least Squares Problems. MSc Thesis, School of Computer Science, McGill University (2011)

  25. Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)

    Article  Google Scholar 

  26. Lenstra Jr, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    Article  Google Scholar 

  27. Mehrotra, S., Li, Z.: Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices. J. Glob. Optim. 49, 623–649 (2011)

    Article  Google Scholar 

  28. Micciancio, D., Voulgaris, P.: Faster exponential time algorithms for the shortest vector problem. In: Proceedings of SODA, pp. 1468–1480 (2010)

  29. Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM J. Comput. 42, 1364–1391 (2013)

    Article  Google Scholar 

  30. Pataki, G., Tural, M., Wong, E.B.: Basis reduction and the complexity of branch-and-bound. In: Proceedings of SODA, pp. 1254–1261 (2010)

  31. Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66, 181–199 (1994)

    Article  Google Scholar 

  32. Schnorr, C.P.: Progress on lll and lattice reduction. In: Nguyen, P.Q., Vallée, B. (eds.) The LLL Algorithm, Information Security and Cryptography, pp. 145–178. Springer, Berlin (2010)

  33. Tan, P., Rasmussen, L.K.: The application of semidefinite programming for detection in CDMA. IEEE J. Sel. Areas Commun. 19, 1442–1449 (2001)

    Article  Google Scholar 

  34. Teunissen, P.J.G., Kleusberg, A.: GPS for Geodesy. Springer, Berlin (1998)

  35. van Emde Boas, P.: Another NP-Complete Partition Problem and the Complexity of Computing Short Vectors in a Lattice. Technical Report Rep. 81–04, Mathematics Institute, Amsterdam, The Netherlands (1981)

  36. Xie, X., Chang, X.W., Al Borno, M.: Partial LLL reduction. In: Proceedings of IEEE GLOBECOM, 5 pp (2011)

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Acknowledgments

We are grateful to two anonymous referees for their detailed criticisms that helped us improve the paper. We also acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Canada Research Chair in Discrete Nonlinear Optimization in Engineering, GERAD, Montreal, QC, H3C 3A7, Canada

    Miguel F. Anjos

  2. École Polytechnique de Montréal, Montreal, QC, H3C 3A7, Canada

    Miguel F. Anjos

  3. School of Computer Science, McGill University, Montreal, QC , H3A 0E9, Canada

    Xiao-Wen Chang

  4. Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 3G8, Canada

    Wen-Yang Ku

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  1. Miguel F. Anjos
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  2. Xiao-Wen Chang
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Correspondence to Miguel F. Anjos.

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Anjos, M.F., Chang, XW. & Ku, WY. Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems. J Glob Optim 59, 227–242 (2014). https://doi.org/10.1007/s10898-014-0148-4

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